Find values for which the limit exists

What I have only found so far is that for all variables non-zero the limit doesn't exist. Anyway, I have no clue how to find the conditions for which it does. I tried a = b = c = 0, but it doesn't seem to help to me...

For functions like this, where you have two variables, I find it best to convert to polar coordinates. That way, exactly one variable, r, measures the distance to (0,0) which is the crucial factor. In polar coordinates,
[itex]x= r cos(\theta)[/itex] and [itex]y= r sin(\theta)[/itex] so
[itex]xy= r^2 cos(\theta)sin(\theta)[/itex], [tex]x2= r^2 cos^2(\theta)[/itex], and [tex]y^2= r^2 sin^2(\theta)[/itex].
Of course, then [itex]ax^2+ bxy+ cy^2= ar^2cos^2(\theta)+ br^2sin(
theta)cos(\theta)+ cr^2sin^2(\theta)[/itex] so that
[itex]ax^2+ bxy+ cy^2= r^2(acos^2(\theta)+ bsin(
theta)cos(\theta)+ csin^2(\theta)[/itex].

Notice that there is no "r" in that! This can have a limit as r-> 0 only if it does NOT depend on [itex]\theta[/itex]- it is a constant. One obvious choice for a,b,c is a= c= 0, b= 1 but there may be others.